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Properties

Label 750.6.c
Level $750$
Weight $6$
Character orbit 750.c
Rep. character $\chi_{750}(499,\cdot)$
Character field $\Q$
Dimension $80$
Newform subspaces $8$
Sturm bound $900$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 750 = 2 \cdot 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 750.c (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(900\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(750, [\chi])\).

Total New Old
Modular forms 770 80 690
Cusp forms 730 80 650
Eisenstein series 40 0 40

Trace form

\( 80 q - 1280 q^{4} - 6480 q^{9} + O(q^{10}) \) \( 80 q - 1280 q^{4} - 6480 q^{9} - 20 q^{11} - 880 q^{14} + 20480 q^{16} - 11340 q^{19} - 1440 q^{21} + 880 q^{26} + 6040 q^{29} - 42300 q^{31} - 880 q^{34} + 103680 q^{36} + 39060 q^{39} + 10740 q^{41} + 320 q^{44} - 1440 q^{46} - 180140 q^{49} - 10440 q^{51} + 14080 q^{56} + 28900 q^{59} + 48880 q^{61} - 327680 q^{64} + 13680 q^{66} - 43560 q^{69} + 9480 q^{71} + 74160 q^{74} + 181440 q^{76} - 215220 q^{79} + 524880 q^{81} + 23040 q^{84} - 10880 q^{86} + 51780 q^{89} + 101760 q^{91} + 26880 q^{94} + 1620 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(750, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
750.6.c.a 750.c 5.b $8$ $120.288$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None 750.6.a.d \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-4\beta _{1}q^{2}-9\beta _{1}q^{3}-2^{4}q^{4}-6^{2}q^{6}+\cdots\)
750.6.c.b 750.c 5.b $8$ $120.288$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None 750.6.a.c \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-4\beta _{4}q^{2}-9\beta _{4}q^{3}-2^{4}q^{4}-6^{2}q^{6}+\cdots\)
750.6.c.c 750.c 5.b $8$ $120.288$ 8.0.\(\cdots\).3 None 750.6.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-4\beta _{1}q^{2}+9\beta _{1}q^{3}-2^{4}q^{4}+6^{2}q^{6}+\cdots\)
750.6.c.d 750.c 5.b $8$ $120.288$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 750.6.a.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+4\beta _{1}q^{2}-9\beta _{1}q^{3}-2^{4}q^{4}+6^{2}q^{6}+\cdots\)
750.6.c.e 750.c 5.b $12$ $120.288$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None 750.6.a.k \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-4\beta _{6}q^{2}-9\beta _{6}q^{3}-2^{4}q^{4}-6^{2}q^{6}+\cdots\)
750.6.c.f 750.c 5.b $12$ $120.288$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None 750.6.a.l \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-4\beta _{6}q^{2}-9\beta _{6}q^{3}-2^{4}q^{4}-6^{2}q^{6}+\cdots\)
750.6.c.g 750.c 5.b $12$ $120.288$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None 750.6.a.i \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-4\beta _{7}q^{2}+9\beta _{7}q^{3}-2^{4}q^{4}+6^{2}q^{6}+\cdots\)
750.6.c.h 750.c 5.b $12$ $120.288$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 750.6.a.j \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-4\beta _{1}q^{2}+9\beta _{1}q^{3}-2^{4}q^{4}+6^{2}q^{6}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(750, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(750, [\chi]) \simeq \) \(S_{6}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(125, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(250, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(375, [\chi])\)\(^{\oplus 2}\)